Although the logical positivists held a wide range of beliefs on many matters, they were all interested in science and skeptical of theology and metaphysics. Early on, most logical positivists believed that all knowledge is based on logical inference from simple "protocol sentences" grounded in observable facts. Many logical positivists supported forms of materialism, philosophical naturalism, and empiricism.
Early critics of logical positivism said that its fundamental tenets could not themselves be formulated in a way that was clearly consistent. The verifiability criterion of meaning did not seem verifiable; but neither was it simply a logical tautology, since it had implications for the practice of science and the empirical truth of other statements. This presented severe problems for the logical consistency of the theory.
It is this grounding in observable facts, and it not being an Axiomatic system, that gives its appeal. When most people I talk to think of "logic" and reason, this is what they are talking about. This is often why I am confounded by people who tell me higher mathematics is not "logical". It is just that maths is not necessarily grounded in observable facts. This is the "Logic" that Dawkins uses, which has been shown to have perpetual consistency difficulties, which should colour the claims that God is inconsistent with observable phenomena. This differs from what I call "logic", which is based on provably consistent axiomatic reasoning. I insist that "reality will not contradict itself" and that axiomatic reasoning is vitally important in Science, and in our understanding of our place in the Universe.